Basic Equations of Micromagnetism

All calculations of the free energy of ferromagnetic crystals require the knowledge of the local magnetic field $\mathbf{H}(\mathbf{r})$ at the point $\mathbf{r}$ . A theory of the local field has been developed, concerning nonlinear changes of $\mathbf{M}$ which may be caused by many distortions of the sample. Within a certain volume, lying symmetrically around $\mathbf{r}$ , the direction $\mathbf{m}(\mathbf{r})$ of the magnetization is given by a Taylor's expansion up to the second derivatives ${\partial }^2{m}_i(\mathbf{r}{\mathrm{)/\partial x}}_i{\mathrm{\partial x}}_k$ at the point $\mathbf{r}$ . The part of the stray field, which is due to the magnetization divergences in this region, is the most important one. It is a sort of coupling field, which is not included in the case of one isolated distortion, which has been solved by Brown. The local field, derived in this way, has been used for the calculation of the free energy. The variational principle leads to a system of differential equations for the determination of $\mathbf{M}(\mathbf{r})$ . These basic equations may be used in all cases of micromagnetic calculations in the many distortion case. In this theory the influence of the intrinsic stray field is of quite another kind than in Brown's approximation. The coupling length of the stray field can be of macroscopic dimensions. It depends not only on the magnetization but also on the applied field and on the kinds of lattice disturbances. In general, the stray field cannot be neglected even in high applied fields.